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Let$\mathbf{v}_{1}=\left[\begin{array}{r}{-1} \\ {-3} \\ {4}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{0} \\ {-3} \\ {1}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{1} \\ {-1} \\ {4}\end{array}\right], \mathbf{v}_{4}=\left[\begin{array}{r}{1} \\ {1} \\ {-2}\end{array}\right]$$\mathbf{p}_{1}=\left[\begin{array}{r}{1} \\ {-1} \\ {2}\end{array}\right], \mathbf{p}_{2}=\left[\begin{array}{r}{0} \\ {-2} \\ {2}\end{array}\right]$and $S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\} .$ Determine whether $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ are inconv $S .$

$p _ { 1 } \notin \operatorname { conv } ( S )$ and $p _ { 2 } \in \operatorname { conv } S$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

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again. One way we can use Thio. See if this points are in a complex. All of is by finding their very central coordinates and by checking that all of them are positive. One way to find their very centric coordinates is by writing H p I minus fee one as a linear combination. Off feed to my newsfeed. One 83 minus 31 Aunt B four minus 51. So we want to find cities three and four on. We can do that by reducing the following no rented matrix. So the first column is gonna be once you row ministry, which is V two minus 31 then 53. Ministry one on before Ministry one on. Then the next column is gonna be p one minus V one on finally p two minus V one. So we're going to leave the first row as it iss on, then 1 to 2 to one on. Then we're going to multiply the second Royal by one half onto the third road. We're going to add three times the first row. Now we're going to subtract one third of the third growth to the free throw. Andi, Also, we're gonna subtract 1/6 off the third road to the second round and we're going to divide the grove by six. Finally, we subtract a second road to the first row on we multiply the second role by one half. From here we can switch the 2nd and 3rd wrote on, We can create the values for C two, C three and C four. Um, we have one third of 50 minus 51 plus two Church fit three minus V one plus 1/6 of before minus 51 equals two p one minus +31 On. When we rewrite this expression, we obtain the following minus 16 to 51 plus 1 30 ft two plus the church of feet three, please. 1/6 off before equals to be one. Andi here since the first party century coordinate its negative. Then we conclude that P one is not in a complex whole. Off is on from the last calendar. For reduce matrix, we have that one third of feet to minus V one plus 1/6 off 53 minutes V one plus 16 before minus. V. One is the same as B two minus V. one on again. If we write this in the following way, one third of feet walk this one third feet too. Plus 1 63 3 plus 1/6 before equals to pick two. We can see that this linear combinations that this linear combination has all of its coefficients, which are positive numbers. Therefore, P two is in the comeback selloff s.

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